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Physics 42200
Waves & Oscillations
Spring 2013 SemesterMatthew Jones
Lecture 24 – Propagation of Light
Midterm Exam:
Date: Wednesday, March 6th
Time: 8:00 – 10:00 pm
Room: PHYS 203
Material: French, chapters 1-8
Reflection and Transmission Coefficients
• In Lecture 19 we derived the relation between the
amplitudes of reflected and transmitted waves:
– Reflection coefficient: � = (�� − ��)/(�� + ��)
– Transmission coefficient: � = 2��/(�� + ��)
– Example: � > � so �� < ��
��
����
• The relation between the power and the amplitude of the wave was:
� =1
2�����
• In the case of the string,
�� = � ��⁄ = � � �� = � ��⁄ = � �
� =�� − ��
�� + ��=
�� − ��
�� + ��
� =2��
�� + ��=
2��
�� + ��
Reflection and Transmission Coefficients
• Let’s check whether this makes sense…
• Suppose the string were attached to an
immovable object.
– This can be approximated by letting → ∞
– Impedance is �� = � � → ∞
– Reflection coefficient for the amplitude:
� =�� − ��
�� + ��→ −1
– Reflected pulse is inverted, as expected.
Reflection and Transmission Coefficients
• In transmission lines:
��
��= −� � � = −���′�(�)
• Reflected pulse: � �, # = � � + �# = �(�)$%&'
��
��=
1
�
��
�#=
��
�� � = −���(�(�)
• Relation between current and voltage of reflected pulse:
� � = −��(� � =�(
�()(� � =
�(
)(� � = −��(�)
• Voltage reflection coefficient should be opposite the current reflection coefficient.
Reflection and Transmission Coefficients
• Current reflection coefficient:
�* =�� − ��
�� + ��
• Voltage reflection coefficient:
�+ =�� − ��
�� + ��
• Transmitted current:
�' = �%� = �% 2��
�� + ��
• Transmitted voltage:
�' = ���' =��
���%
2��
�� + ��= �%
2��
�� + ��
Reflection and Transmission Coefficients
• Does this make sense?
• Reflection from a shorted transmission line: �� = 0
• Voltage across the wire at the end must be zero:
incident and reflected voltages cancel
�+ =�� − ��
�� + ��= −1
Reflection and Transmission Coefficients
• Does this make sense?
• Reflection from an open transmission line: �� = ∞
• Voltage across the wire at the end must be zero:
incident and reflected voltages cancel
�+ =�� − ��
�� + ��= +1
Reflection and Transmission Coefficients
• Does this make sense?
• Reflection from a shorted transmission line: �� = 0
• Current across the wire at the end should be
maximal: incident and reflected currents don’t cancel
�* =�� − ��
�� + ��= +1
Reflection and Transmission Coefficients
• Does this make sense?
• Reflection from an open transmission line: �� = ∞
• Current across the wire at the end should be zero:
incident and reflected currents must cancel
�* =�� − ��
�� + ��= −1
Reflection and Transmission Coefficients
Geometric Optics
• Wave equation in free space:
��-.
�/�= 010
��-.
�#�=
1
��
��-.
�#�
• Impedance of free space: �0 = 02 = 377Ω
• Free space is isotropic, so � = 0 and � = 1
• Waves propagate in the forward direction
according to the wave equation.
• What about wave propagation in transparent
media?
Geometric Optics
• Huygens’ principle: Light is continuously re-emitted in all
directions from all points on a wave front
– Further developed by Fresnel and Kirchhoff
– Destructive interference except in the forward direction
Forward Propagation
• Remember the forced harmonic oscillator problem:
6�7 + 8�9 + :� = ;0$%&'
� =;0 6⁄
�0� − �� �
+ �< �
= = tanA��<
�0� − ��
• When the driving force is much greater than the resonant frequency, = → 180°
• Electrons bound to molecules act as tiny oscillators that are driven by the electric field.
Forward Propagation
At point P the scattered waves are more or
less in-phase:
• constructive interference of wavelets
scattered in forward direction.
• Destructive interference of wavelets and
incident light in the backward direction.
True for low and high density substance
Forward Propagation
• At lower frequencies the
phase shift is less than 180°
• The phase lag results in the
scattered waves having a
shorter wavelength
D( =�
E<
2
E
• Index of refraction:
F =2
�• Applies to the phase velocity
Reflection
• Energy is reflected from an interface between
materials with different indices of refraction
• Reflection coefficient:
� =�� − ��
�� + ��=
1 F�⁄ − 1/F�
1 F�⁄ + 1/F�=
F� − F�
F� + F�
n1 > n2n1 < n2
External reflection:
180° phase shift
Internal reflection:
No phase shift
Reflection: Microscopic View
When an incident plane wave front
strikes the surface at some angle it does
not reach all the atoms along the
surface simultaneously.
Each consequent atom will scatter at
slightly different phase, while the
spherical wave created by previous
atom had a chance to move away some
distance.
The resulting reflected wave front
created as a superposition of all
scattered wavelets will emerge also at
an angle to the surface.
Reflection: Constructive Interference
For constructive interference
spherical waves created by
the atoms on the surface
must arrive in-phase.
Consider two atoms on the
surface.
Wave function depends only on : ∙ HI − �#: - = E(: ∙ HI − �#)
Wave phase along the incident wavefront is the same.
Scattered wavefront: Points C and D must also have the same phase.
)(
)(
tBDkfE
tACkfE
D
C
ωξωξ
−+⋅=
−⋅+=
phase shift due to scattering atom
BDAC =
The Angle of Reflection
o90=∠=∠=
=
CB
ADAD
BDAC
Triangles ABD and ACD:
The Law of Reflection (1st part):
The angle-of-incidence equals the angle-of-reflection:
JK = J%
Rays and the Law of Reflection
A ray is a line drawn in space along the
direction of flow of the radiant energy.
Rays are straight and they are
perpendicular to the wavefront
Conventionally we talk about rays instead
of wavefronts
The Law of Reflection
1. The angle-of-incidence equals the angle-of-reflection JK = J%
2. The incident ray, the perpendicular to the surface and the
reflected ray all lie in a plane (plane-of-incidence)
Specular and Diffuse Reflection
Smooth surface:
specular reflection
Rough surface:
diffuse reflection
Refraction
• The speed changes in different media, but the
frequency remains the same
– Wavelength is shorter in the “slow” medium:
D =�
E=
2
FE
• The phases must be constant everywhere on a wave
front:
D�
D�
F�
F�
Refraction
• When light enters a medium with a larger index of refraction, it bends towards the direction that is normal to the surface.
D�
D�
J�
J�
�
� cos 90P − J� = � sin J� = D�
� cos 90P − J� = � sin J� = D�
� =D�
sin J�=
D�
sin J�
D =2
FE2
E
1
F� sin J�=
2
E
1
F� sin J�
F� sin J� = F� sin J�
Total Internal Reflection
• What happens when J� ≥ 90P?
J�
J�
F�
F�
90P
JS
F�
F�
J�( = J�J�
F�
F�
Critical angle:
F� sin JS = F�
sin JS =F�
F�